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PACIFIC JOURNAL OF MATHEMATICSVol. 102, No. 1, 1982

ON THE NUMBER OF REAL ROOTSOF POLYNOMIALS

THOMAS CRAVEN AND GEORGE CSORDAS

A general theorem concerning the structure of a certainreal algebraic curve is proved. Consequences of this theoreminclude major extensions of classical theorems of Pόlya andSchur on the reality of roots of polynomials.

1* Introduction and definitions* Throughout this paper, allpolynomials are assumed to have real coefficients. In 1916, GeorgePόlya wrote a paper [6] in which he considered two polynomials,/(«) = ΣΓ=o diX*, am Φ 0, and h{x) — Σ?=o MS K Φ 0, where n ;> m,both polynomials have only real roots and the roots of h areall negative. Under these hypotheses, he proved the followingtheorem.

THEOREM 1.1 (Pόlya [6]). The real algebraic curve Fix, y) =hf(y) + b&f'iy) + + bmxmf{m){y) — 0 has m intersection pointswith each line sx — ty + u = 0, where s ^ O , £ ̂ 0, s + ί > 0 andu is real*

Pόlya then noted that this theorem gives a unified proof ofthree important special cases regarding composite polynomials:

(1.2)(a). Setting x = 1 gives a special case of the Hermite-Poulain theorem [5, Satz 3.1].

(1.2)(b). Setting y = 0 gives a theorem of Schur [5, Satz 7.4].(1.2)(c). Setting x — y gives a theorem of Schur and Pόlya [7,

p. 107].

Our main theorem, proved in § 2 establishes some of the prop-erties of F(x, y) = 0 when we drop all restrictions on / and requireonly that all the roots of h be real (of arbitrary sign). In thelast section, we apply this theorem to obtain an extension ofTheorem 1.1 which states that, for h restricted as in Pόlya's theorem,there are at least as many intersection points as the number ofreal roots of /. As corollaries, we then obtain the full strengthof the Hermite-Poulain theorem and extensions of the theorems ofPόlya and Schur to arbitrary polynomials /. Pόlya states that histheorem is one of the most general known theorems on the realityof roots of polynomials. We believe this statement is still true

15

16 THOMAS CRAVEN AND GEORGE CSORDAS

today, more than sixty years later, especially for our much moregeneral version of it. The generalizations that this provides of(1.2)(b) and (c) are used in [2] to solve an open problem of Karlinon characterizing zero-diminishing transformations which he says"seems to be very difficult." They also are used in [1] for manyfurther applications to polynomials and entire functions. For theknown results on theorems of this type, the reader is referred to[4] and [5].

Because the subject of algebraic geometry deals almost exclu-sively with algebraically closed fields, we have found it necessaryto develop some new terminology. As much as possible, we followWalker's book "Algebraic Curves" [8]. In particular, we use theword component to refer to an irreducible factor of the curve F(x, y).Since we deal only with the real points of a (not necessarily irredu-cible) algebraic variety, we need a term to refer to the individualparts of the curve F(x, y) = 0, even though these parts may notbe components, or even connected components since two parts mayintersect. For this we shall use the word branch. Classically,branches are only defined in a neighborhood of a point on the curve[3, p. 39] where they have the usual analytic meaning. We shalluse the word branch in the following global sense: the branch ofthe curve F containing a given point on the curve is obtained bytravelling along the curve in both directions until reaching asingularity or returning to the starting point; at a singularitytravel out along the other arc of the same local branch on whichyou arrived. Thus the branches are the "pieces" into which acircuit in the protective plane [3, p. 50] is broken by removing theline at infinity. We shall see in the next section that the branchesin which we are most interested will always go to infinity in bothdirections. In particular, if / has simple roots, the curve F(x, y)in Theorem 1.1 is nonsingular and the branches are just the con-nected components. As is usual in algebraic geometry, all roots,branches and components will be counted including multiplicities,with the exception of Lemma 2.1.

2* The main theorem* We begin with a lemma which describesthe variety F(x, y) = 0 when the polynomial h(x) has degree one.It will be used in an inductive step in the proof of the maintheorem.

LEMMA 2.1. Let f(y) = Σ? = o

α<lΛ an =£ 0> be any polynomialwith real coefficients and let a be any nonzero real number. SetF{x, y) = af(y) + xf'(y). Assume that f has r real roots alf , ar

and that f has s real roots βu •• ,/3β, listed with multiplicity if

ON THE NUMBER OF REAL ROOTS OF POLYNOMIALS 17

and only if they are also roots off. Then the real variety F(x, y) = 0satisfies the following;

(1) The variety consists ofs + 1 one-dimensional branches(not necessarily distinct), exactly r of which cross the y-axis.

( 2 ) The branches are disjoint unless f has a multiple realroot. At a root at of multiplicity k, we have k — 1 componentscoinciding with the line y = at and one other branch through (0, at).Other than coincidental lines y = aif branches can only intersect onthe y-axis.

(3) Each branch which intersects the y-axis will intersect itonly once and will also intersect each line x = c, where c is a con-stant.

(4) All branches are asymptotic to straight lines from the sety = βif i — 1, 2, , s and y — —na~ιx — an_xa~ι.

Proof. Since F(x, y) — af(y) + xf\y) = 0, we can write

(2.2) x = - U . 2 M whenever f(y) Φ 0 .

If / has a root at of multiplicity k, then F(x, y) has a factor of(y — oίiY'1. Canceling this from the numerator and denominator in(2.2) shows that there is still a branch passing through (0, at).Other than this situation, it is clear from (2.2) that the branchesare all disjoint so that claim (2) of the lemma holds. From (2.2)we also see that for large values of y, the graph is asymptotic tothe line y = — na~λx — an_xa~λ. Thus this line together with thelines y = β< (arising from f'(βt) — 0) form the set of asymptotes andclaim (4) holds. Since there are s points βif claim (1) will followfrom the arguments above and (3). Now consider the interval (y19

y2) where yγ and y2 are two consecutive roots of / ' which are notroots of /, or y1 — — oo if / ' has no such root less than y2, or y2 —oo if / ' has no such root greater than yle To prove claim (3), weneed only consider the case where there exists a point y0 in (yl9 y2)such that f(y0) = 0. Note that there can be at most one such pointby the choice of yx and y2 and Rolle's theorem. Assume first thatf'(yQ) Φ 0. Then / ' has constant sign on (yu y2), so that / is mono-tonic on the interval. Since y0 is a simple root of /, equation (2.2)implies that x ranges from — oo to oo (or vice versa) as y rangesfrom yx to y2. Thus this branch intersects each line x — c wherec is an arbitrary constant. Now assume that y0 is a root of / ofmultiplicity k. The k — 1 components y — y0 satisfy (3). The re-maining branch through (0, yQ) behaves as in the case where f'(y0)is nonzero since f\y) can be factored as (y — yQ)k~~ι times a function


which has constant sign on (ylf y2) and f(y) factors as (y — y^)k~ι

times a polynomial which has a simple root at y0. Since / is mono-tonic on each subinterval (yl9 y0) and (yQ, y2), equation (2.2) impliesthat for this branch, x ranges from -oo to ^ (or vice versa) asy ranges from yx to y2. This concludes the proof of the lemma.

We now proceed to our main result. The main point of thistheorem is to guarantee that no branch of F(x, y) = 0 can passthrough two distinct roots of f(y) on the y-axis.

THEOREM 2.3. Given h{x) = Σ?=o hx*, n^l, boφ 0, bn = 1, withonly real roots and f(y) an arbitrary polynomial, form F(x, y) =Σ f ^ o W 1 ^ ) ' Each branch of the real curve F{x, y) — 0 whichintersects the y-axis will intersect the y-axis in exactly one pointand will intersect each vertical line x — c, where c is an arbitraryconstant. If b0 = 0, the conclusion still holds for all branchs whichdo not coincide with the y-axis. Furthermore, if two brancheswhich cross the y-axis intersect at a singular point (x0, y0) not onthe y-axis, then these branches are in fact components of the formy — y0 = 0, and thus coincide as horizontal lines.

Proof The theorem follows from the following somewhatstronger statement which we shall prove:

(2.4)(a). In the graph of F(x, y) = 0, there does not exist apair of disjoint paths from the y-axis to any point (x0, y0) off they-axis; (We say two paths are disjoint if their interiors have nopoint in common.)

(2.4)(b). Two coincidental paths from the y-axis to a point(x0, y0) with x0 Φ 0 are necessarily horizontal lines y = yo;

(2.4)(c). Multiple straight line components can intersect nobranches off the y-axis that are not coincidental straight lines.

If b0 — 0, let br, r > 0, be the first nonzero coefficient in h andwrite g{y) = fr\y). Then F(x, y) = xrΣΛΐ=obi+rx

ίg{ί)(y) has r compo-nents which coincide with the y-axis and the new polynomials whichreplace h and / satisfy the hypotheses of the theorem. Thus wemay assume b0 Φ 0.

Note that it suffices to prove statement (2.4) for points withx0 > 0. For if we replace h(x) by h( — x), these points are reflectedabout the y-axis and the new polynomial h will still have only realroots.


We now prove the theorem by induction on n, the degree ofh. If n = 1, the conditions of (2.4) follow from Lemma 2.1. Nowassume n > 1 and that (2.4) holds for polynomials h of degree lessthan n and arbitrary polynomials /. Assume now that h has degreen. Since h has only real roots, we can factor h(x) = (x + aQ(x + an). Let α = αn and g(x) = (x + αx) -(α? + αΛ_i), so that fc(a?) =(a? + a)g(x). Write g(x) = ΣS=o dtx

im, the induction hypothesis saysthat the real algebraic curve

Fîx, y) = dof(v) + dxxf{y) + - + d^x^ψîy) = 0

satisfies statement (2.4).Next we define a variety in three variables by

(2.5) G{x, yy t) = αίVΛί, y) + xJ-F^t, y) = 0 .

For any fixed value of ί = to> we can apply Lemma 2.1 to G(x, ?/, to)where the polynomial in question is F^Û, y). In particular, theintersection of each plane t — t0 with the variety G = 0 satisfiescondition (2.4) with the i/-axis replaced by the line x = 0 = t — t0.We shall refer to this algebraic curve as Gt(x, y) = 0. For any fixedvalue of x — x0, we obtain

, ί) = aFn_γ{t, y) + x^F^t, y)oy

Σidtt^Ύι=o dy

By applying the induction hypothesis to g(t) and the polynomial<xf(y) + Xof\y)f we see that the intersection of each plane x = x0

with the variety G = 0 also satisfies condition (2.4) in the variables£ and y with the #-axis replaced by the line x — x0 = 0 = t. Weshall refer to this curve as Fn_ltXo(t, y) = 0. Finally, we considerthe intersection of (? = 0 with the plane x = t, and note that astraightforward computation shows that G(x, y, x) — Fn{x, y) for allvalues of x and y, where Fn(x, y) = F(x, y) = 0 is the curve involv-ing the polynomial fc(α?) that the statement of the theorem is con-cerned with. Thus it will suffice to show that the curve obtainedfrom the intersection of G — 0 with the plane x — t satisfies condi-tion (2.4). We identify the curve in the plane x = t with thevariety Fn = 0 and the curve Fn_1>0 in the ^/ί-plane with the curveFn_x = 0. For the remainder of this proof, we shall write 'Vaxis"in quotes to mean the appropriate line in any of the above planeswith x or t constant, for which we know that the curve inducedby G = 0 satisfies condition (2.4).


Now we assume that condition (2.4)(a) is violated by the curveFn = 0, but (2.4)(b) and (2.4)(c) both hold. Choose paths Pt and Pz

in the x = t plane which give the contradiction to the theorem insuch a way that the area enclosed by the paths and the ?/-axis isminimal; in particular, the points yx *z y2 where the paths intersectthe y-axis are consecutive roots of /. We also allow the possibilitythat the paths begin at a multiple root y1 = y2 of / with the pathPx through 2/1 having greater y~values for x near zero than the pathP2 through y2. We may also assume that (x0, y0) is a point forwhich x0 is the maximum value taken on by x in both paths.

We shall consider the algebraic curve Gt = 0 for each fixed t.The points of the paths Pi are obtained from the intersection ofthe branches of Gt with the plane x = t. Unfortunately, thesebranches do not always vary continuously with t. The followinglemma describes how discontinuities can occur.

LEMMA 2.7. Set ft(y) = Fn_x(t, y), so that for fixed t equation(2.5) becomes

(2.8) Gt(x, y) = aft(y) + xft(y) .

In Gt(x, y) — 0, the 'partition of the points of the curve into branchesfor fixed t varies continuously with t except possibly at a pointt = s such that fs or // has a multiple real root.

(a) // fs has a double real root at y0, then for values of tnear s, the curve Gt(x9 y) may have two branches crossing the y-axisnear y0 or may have two branches coming close to (0, yQ) and con-tained entirely on opposite sides of the y-axis. If f8 has a root yQ

of multiplicity m > 2, then for values of t near s, the curve Gt

has r ^ m branches crossing the y-axis near yQ; the number m — ris even; and there are at most (m — r)/2 branches contained entirelyin the half plane x > 0.

(b) If fs has a real root of multiplicity m > 1 at y0 withfs(Vo) =£ 0, then the curve G8 has a (multiple) horizontal asymptotey = y0. For nearby values of t, the asymptotes can separate, creat-ing new branches (not crossing the y-axis) or disappear in pairsf

causing adjacent branches which were separated by an even numberof asymptotes to join into a single branch. (We shall refer to thisas ζ<joining at infinity.'9) Note that f[ must have an odd numberof roots between any two consecutive roots of ft, so two branchescrossing the y-axis cannot join at infinity.

Proof. The lemma follows almost immediately from Lemma 2.1and Rolle's theorem. By Lemma 2.1, two branches can only meet


at the i/-axis, corresponding to a multiple root of ft, or at infinity,corresponding to a multiple asymptote of even multiplicity (i.e., amultiple real root of //). Thus discontinuities can only occur incases (a) and (b). The analysis of the individual cases is clear fromLemma 2.1 and a consideration of how multiple roots of a poly-nomial and its derivative may vary as the polynomial varies con-tinuously.

We now continue the proof of the theorem. Assume first thatVι = 2/2 and, as t increases, the branches of Gt which generate thepaths in Fn trace out straight lines y = y1 = y2 in Fn_±. For thetwo paths P1 and P2 to meet at (x0, y0, x0), the corresponding branchesof Gt must vary to become a pair of straight lines y = yQ = yι inGχQ(%, V) = 0; but then Lemma 2.1 implies that there is at least onemore branch of GH passing through (0, y0, x0). Thus the straightlines in Fn_x meet another branch at (0, y0, xQ), in contradiction to(2.4)(c). Therefore, using (2.4)(b) and (c), we can conclude that ast increases from zero, the branches of Go through y1 = y2 mustseparate, continuing to cross the 2/-axis, for small values of t, atthe points where the branches of Fn_x through yx = y2 meet theconstant t planes.

As we analyze the behavior of the branches of Gt, we shall belooking for a contradiction in one of the curves Fn_1>c where c isa constant with 0 ̂ c <^ xQ. The branches of Go passing through yx

and y2 give points (c, y^c), 0) in the plane x = c for i = 1, 2, respec-tively. If there is more than one choice for ^(c), let y^c) denotethe set of y-values in the plane x = c corresponding to y^

From equation (2.5) and the induction hypothesis on Fn_ιy wesee that the branches through yx and y2 in Go cannot continuouslyvary to join (or rejoin if yx = y2) each other at the "y-a,xis" as tincreases, for this would create a path in Fn_x contradicting theinduction hypothesis. By Lemma 2.7(b), they also cannot join atinfinity since there are an odd number of roots of / ' between yι

and y2 by Rolle's theorem. On the other hand, they must somehowvary so that the two paths in Fn meet at (xOf y0, x0), which mustbe a point of GXQ at which (d/dy)GXQ = 0 . By Lemma 2.1, this pointwill either lie on a single branch of GXQ or will lie on a set ofmultiple straight lines y = y0 (in the plane t = x0). Thus thebranches of Go through yt and y2 must first join with other branches(which did not cross the #-axis at t — 0 or else this would againlead to a contradiction in Fn_^).

The analysis of what happens using Lemma 2.7, especially withregard to behavior at infinity, requires that we know the sign ofa, which determines the slopes of the branches of Gt at the "y-


axis." Henceforth, we shall assume a > 0. The analysis for a < 0is similar and will be omitted. Since a > 0, branches can only bejoined at infinity if the lower one does not cross the ^/-axis. Ifsuch a branch lies between the branch through yx and the branchthrough y2, then joining it to the branch through yι has essentiallyno effect, since the branch through yx could have been continuouslyvaried to obtain the same result. So let (0, y, t0) be the point onthe "y-axis" at which two branches of Gt come to the "?/-aχis," orappear as a pair of straight lines through a root of ft as in Lemma2.7(a), prior to at least one of them joining with one of our originalbranches. Let yt(t) denote the ̂ /-coordinate at x = 0 of the branchof Gt continuously obtained from the original branch through' yt\this is well defined for t > 0 until the first time y^t) is a multipleroot of ft. Then we have three possibilities:

(A) y,{Q > y > y2(t0)

(B) y ^ y,{Q

(C) y £ y2(Q .

We consider the effect of each of these possibilities and showthat no combination of them can allow the paths in Fn to occurwithout giving rise to a contradiction in one of the curves Fn_Uc,where 0 ̂ c ̂ x0, which, as we have seen, must satisfy the induc-tion hypothesis.

Assume first that situation (A) occurs alone. Then the onlyway to have the paths Px and P2 meet each other is to have oneof the new real roots of ft vary to meet yt(t) and the other tomeet y2(t), and then subsequently joining these branches. (Notethat connections at infinity cannot be used since they require thatat least one branch being joined not cross the i/-axis. Also thepolynomial // has an odd number of roots between y^t) and j/2(t) )But we obtain a contradiction as soon as both of the new brancheshave joined to the original ones since this generates a path in Fn_x

from (0, yx) to (0, y2) passing through the point (<0, y). Indeed, morethan one application of (A) can do no better; to avoid leaving aline between the branches and thus separating P1 and P2, the newbranches all must eventually join (prior to or at t = x0) to inducea contradiction in the curve Fn_lt

No branch can connect via infinity with the branch throughy^t) since a > 0 (see equation (2.2)). Now we consider the effectof situation (B). The new branch must meet yt(t) and pull off the2/-axis as t increases. The new branch β thus formed cannot joinat infinity with the branch coming from y2 (again because a > 0),hence to obtain the connection of Pί and P2, the branches must


rejoin the "^/-axis" and part again. The lower one must then joineither with the branch through y2(t) or with the upper branch fromsituation (A), while the lower branch joins with the branch throughy2(t). (As noted above, applications of (A) have no substantial effecton the argument.) This use of (B) prevents a contradiction fromoccuring in Fn^ by breaking the path which would otherwise begenerated from yx to y2. But, if we let xm be the maximum valueof x to which the branch β pulls away from the τ/-axis, say att — tm, and note that xm <^ x0 since a point is being generated onPλ by this branch, then the contradiction is to be found in theplane x = xm. Indeed, in the curve Fn_UXm, a path will be gener-ated between the smallest value of y^xj) and the largest value ofy2{xm) on the "y-a,xis" since in this plane it is not broken, but hasa singularity at t = tm. (The only way this could be avoided isby having some other path joining two points of y^x*) or twopoints of y2{xm), which again would contradict the induction hy-pothesis.) If more than one application of situation (B) occurs,the largest value of xm determines the plane in which a contradictionoccurs. Thus situation (B) has virtually no effect other than toshift the plane in which a path is generated which violates theinduction hypothesis.

Situation (C) leads to a contradiction also as for (B) (or anycombination of (A), (B), and (C)) if the branches which leave theτ/-axis later return to it as they had to for (B). The other possi-bility is that they do not return to the ?/-axis, but rather connectwith a branch above (i.e., larger y-values: either a branch throughy^it) or one coming from applying (A) or (B)) by joining at infinity.In this case the contradiction always occurs in Fn_1}Xo since, if noother contradiction occurs with the points of yi(x0), i = 1,2, thenthe paths through the smallest value of yx{xQ) and the largest valueof y2(x0) will meet at (xOf y0, x0). The meeting occurs either as thebranch in question pulls back from infinity to x0 (i.e., the point atwhich (d/dy)Gt — 0 on the branch is the point fe, y0) when t = a?0);or, if this happens for t < x0, then necessarily (A) has occured andanother point of Gt with (β/dy)Gt = 0 will be the last to pull backto x = x0 at t — x0. Note that if a < 0, the arguments for cases(B) and (C) must be reversed. This completes the proof of (2.4)(a).

Next we prove (2.4)(b) and (2.4)(c) in a similar manner. For(2.4)(b), assume that Fn has two coincidental paths from a point y0

on the 2/-axis. Then for each t > 0, there exists a pair of straightlines in Gt which generate the path. These pairs of straight linesgenerate coincidental paths in Fn_lf so by the induction hypothesisthe paths in Fn^ are horizontal straight lines. But then we musthave straight lines in Fn also. Finally, for (2.4)(c), we assume that


Fn has m multiple straight lines y = yQ which intersect some otherbranch of Fn at t = tQ > 0. Then, as above, we obtain m straightlines y = y0 in each Gt, 0 <̂ £ < ίo> and at least m + 1 lines in G v

By Lemma 2.1(2), these generate m + 1 straight lines y = y0 in.FV L which intersect another branch of F ^ at (ί0, yo)f contradictingthe induction hypothesis. This completes the proof of the theorem.

REMARK 2.9. All of the complications in the preceding proofare caused by the fact that it is possible to have branches in Fn_t

which cross the y-axis and have more than one point with the sameίc-coordinate. This sometimes happens when the degree of h is lessthan the degree of /. We conjecture that this does not happen ifthe degree of h is greater than or equal to the degree of /; itcertainly cannot happen if / has only real roots, for then a verticalline would intersect the curve in more points than the degree. Aninteresting example in this regard is obtained by taking h(x) to be apower of x + 1 and f(y) = 11 - 16y + 8y2 - Ayz + y* (see Figure 2.1).Then using the notation of the proof above, the curve F2(x, y) has abranch which intersects some vertical lines in three points, but noother Fn has such a branch. The curve F 4 is also interesting; itcomes very close to having a singularity near (.6, -1.5) causing oneof the branches to intersect some horizontal lines in three points.

V

Fj (X. T) = 0

F4tx, γ) = o

F5(X,Y)

FIGURE 2.1.


Also of interest is the fact that, in going from F2 to F5, the numberof branches increases from two to four.

3* Applications* We begin with a direct extension of Theorem1.1 of Pόlya to arbitrary polynomials /.

THEOREM 3.1. Let h(x) = Σ?=o M* be a polynomial of degree nwith only real negative roots and f{y) an arbitrary polynomialwith r real roots and degree at most n. The real algebraic curveF(x, y) = Σ?=o &i^ί/ίί)(l/) = 0 has at least r intersection points witheach line sx — ty + u = 0 where s ^ 0, t ^ 0, s + t > 0 and u isreal.

Proof. Write f(y) — ΣΓ=o (&<#*, am Φ 0, m ^ n, and set y = txin the equation for the curve F(x, y) = 0; compute

a'1 Km x~mF(x} tx) = bQtm + mbλtm-1 + m(m - l)b2t

m-2

(3.2) | |-~+ ... +m!6»

We claim that this polynomial has only real negative roots. Byhypothesis h(t) = Σ 6ίί

ί has only real negative roots, and therefore

g(t) = bQ + bj

has only real negative roots by [4, p, 343]. But then tmg(m/t),which equals the polynomial in (3.2), has only real negative roots.This implies that each branch of F(x, y) — 0 is asymptotic to lineswith negative slopes, the possible slopes being given by the valuesof t which make the expression in (3.2) equal to zero. For eachof the r real roots of /, Theorem 2.3 implies that there is a branchof F(xf y) which passes through the root on the /̂-axis and forwhich x goes from — oo to + °° as y goes from + oo to — oo sincethe asymptotes have negative slopes. Thus by continuity each ofthe branches must intersect every line with zero, positive or infiniteslope.

For many applications, we prefer to relax the conditions onh{x) slightly and lose some lines in the conclusion. Note that weremove the restriction on the degree of /.

THEOREM 3.3. Let h(x) — Σ?=o hx1 be a polynomial of degree nwith only real nonpositive roots and f{y) an arbitrary polynomialwith r real roots. The real algebraic curve F(x, y) = Σ?=oMί/(ί)(i/) =0 has at least r intersection points with every line of positive slope.


Proof. Assume first that zero is not a root of h. Then theonly difference in hypotheses from the previous theorem is that thedegree of / may be greater than the degree of h. The proof isthus the same as the previous proof, except that the polynomial inequation (3.2) may have zero as a root. That is, the branches ofF(xf y) may have horizontal asymptotes. Hence lines of zero slopemay miss the branches, but lines of positive slope will still intersectany branch which crosses the #-axis.

If zero is a root of h, let bs, s > 0, be the first nonzero coeffi-cient of h and write g(y) = fω(y). Note that g has at least r — sreal roots by Rollers theorem. Then F(x, y) - α?βΣ?=o bt+tfg^ίy)has s components coinciding with the y-B.xi& and the remainder ofthe curve has at least r — s intersections with each line of positiveslope. Since every line of positive slope must cross the y-axia, thetheorem holds.

We can now obtain the corollaries mentioned in the introduction.The first is a corollary of Theorem 2.3.

COROLLARY 3.4 (Hermite-Poulain). Let f{x) be a polynomialwith real coefficients and h{x) = b0 + bλx + + bnx

n a polynomialwith only real roots. Then the polynomial g(x) = Σ?=o &*/(i) (χ) hasat least as many real roots as f. If f has only real roots, thenevery multiple root of g is also a multiple root of f.

Proof. The real curve F(x, y) = Σ?=o MV { i> (2/) = 0 has a branchcrossing the /̂-axis for each real root of /. By Theorem 2.3, thesebranches intersect the line x = 1. The polynomial Σ?=oδi/(<)(i/) hasa root for each intersection, and therefore has at least as manyreal roots as /. If / has only real roots, then every branch ofF(x, y) intersects the τ/-axis. If g has a multiple root c, then F(x,y) has a singular point at (1, c) where two branches intersect. Bythe last assertion of Theorem 2.3, these branches are horizontallines, so that c is also a multiple root of /.

COROLLARY 3.5. Let h{x) = Σ?=o b^ be a polynomial of degreen with only real roots, all of the same sign or zero, and let f(x) =ΣS=otti«S m ^ n. Then the polynomial ΣίUίlαAίc* has at least asmany real roots as f.

Proof. If the roots of h are all negative, we set y = 0 inTheorem 3.1 and note that f{i)(0) = i!α,. If the roots of h are allpositive, replace h{x) by h(—x), apply the theorem, and then switchthe variable again. If h(x) = x8 Σ?=o8 bi+8x\ bs Φ 0, then let g{y) =


fs)(y) and apply Theorem 3.1 to Σ b^x'g^iy), setting y = 0. ByRolle's theorem, the number of real roots of / is at most equal tos plus the number of real roots of g. Thus the conclusion againholds.

A slight change in the hypotheses of this corollary leads us toCorollary 3.6 in which we obtain the result separately for thepositive and negative roots. In [2], Corollary 3.6 is used to solvean open problem of Karlin.

COROLLARY 3.6. Let h(x) — Σ?=o bxx* be a polynomial of degreen with only real negative roots and let f(x)==^T=oaix

i

f m <> n.Then the polynomial ΣϊUϋαAff* has at least as many positive rootsas f has positive roots and at least as many negative roots as fhas negative roots. The multiplicity of zero as a root is the same.

Proof. This follows from the proof of Theorem 3.1; since theasymptotes all have negative slopes, any branch which crosses theτ/-axis at a positive root of / will cross the cc-axis at a positiveroot of ^Σiilaibix

i

f and similarly for negative roots.

COROLLARY 3.7. Let h(x) = Σ?=o &«#* be a polynomial with onlyreal nonpositive roots and let f be an arbitrary polynomial withreal coefficients. Then the polynomial Σ?=o b^f^ζx) has at least asmany real roots as f.

Proof. Set y = x in Theorem 3.3.

This corollary is the starting point for our work in [1], whereit is used to obtain many results concerning polynomials and entirefunctions.

REFERENCES

1. T. Craven and G. Csordas, An inequality for the distribution of zeros of polyno-mials and entire functions, Pacific J. Math., 95 (1981), 263-280.2. , Zero-diminishing linear transformations, Proc. Amer. Math. Soc, 80(1980), 544-546.3. J. Coolidge, A Treatise on Algebraic Plane Curves, Dover, New York, 1959.4. B. Ja. Levin, Distribution of zeros of entire functions, Amer. Math. Soc. TransL,Providence, R.L, 1964.5. N. Obreschkoff, Verteilung und Berechnung der Nullstellsn Reeler Polynome, VebDeutseher Verlag der Wissenschaften, Berlin, 1963.6. G. Pόlya, Uber algebraische Gleichungen mit nur reelen Wurzeln, Vierteljεchr.Naturforsch. Ges. Zurich, 6 1 (1916), 546-548.7. G. Pόlya and J. Schur. Uber zwei Arten von Faktorenfolgen in der Theorie deralgebraischen Gleichungen, J. Reine Angew. Math., 144 (1914), 89-113.


8. R. Walker, Algebraic Curves, Dover, New York, 1950.

Received May 19, 1981. The first auther was partially supported by NSF grantMCS79-00318.

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